The sphere in the diagram has radius \(r\) and is cut by two parallel planes distance \(h\) apart.
One of these planes is at a distance \(x\) from the “north pole” of the sphere, as shown in the diagram.
What is the surface area of the part of the sphere lying between these two parallel planes?
Perhaps we could try to find some exact or approximate answers in certain special cases to get us started – that might give us some insight into the problem. What values could we pick for \(h\) and/or \(x\) which would make the problem either simpler, or allow us to give a reasonable approximate answer?